Classical Wigner approximation

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Semi-classical approximations. In classical formulae, quantum effects may be accounted for to low order. For example, the the Wigner-Kirkwood expansion of the pair distribution function may be used [136, 302],... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Its time evolution is given by full quantum mechanics in the Wigner representation. In order to obtain a computationally tractable form, we consider a limit where the time evolution of W X, X2,t) is approximated by quantum-classical dynamics. 

At somewhat lower temperatures, when quantum effects start to play a role, the basic TST expression (3.29) can be improved in several ways First, classical partition functions 2r and can be replaced by their exact quantum analogues for a harmonic potential. (In fact, this is usually done automatically in TST.) Second, quantum tunneling can be included approximately via the Wigner tunneling correction (3.7) ... 

If information on the reaction path is available, as, for instance, in variational transition state theory, this can be used to calculate k [69,70]. In transition state theory, only the knowledge of the energy and its first and second derivatives at the reactant and transition state locations is needed and the barrier is typically approximated by a simple functional form. One possibility is to describe the reaction barrier by an Eckart potential [75] (also called a sech potential, depending on the literature source), k in Eq. (7.19) is defined as the ratio of transmitted quantum particles to classical particles and the resulting integral for the Eckart potential can be solved numerically. An approximate solution is the Wigner tunneling correction ... 

In the sum over states formula, excited states for the vibrational modes need to be included up to convergence. A more convenient integral expression is provided by classical or semiclassical theories. At high temperatures and low frequencies, the vibrational motion behaves increasingly classically and the semiclassical Wigner-Kirkwood expression is an excellent approximation to the quantum partition function [78] for low-frequency vibrations at pyrolysis temperatures. The semiclassical... 

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